1. Field of the Invention
The invention is related to the field of interpretation of data from well logging instruments comprising a plurality of exponentials. More specifically, the invention is related to methods of processing measurements from a nuclear magnetic resonance logging instrument. By way of example, the method of the present invention is described in relation to the processing of measurements made by a pulsed neutron well logging instrument for determining various properties of earth formations penetrated by a wellbore. The processing method enables separating various components of the signals from the instruments to obtain parameters of interest.
2. Background of the Art
Pulsed neutron well logging instruments are known in the art for determining the macroscopic thermal neutron capture cross-section of earth formations penetrated by a wellbore. A typical pulsed neutron well logging instrument is described, for example, in a sales brochure entitled PDK-100, Western Atlas Logging Services, Houston, Tex. (1994). Typical pulsed neutron instruments include a controllable source of high-energy neutrons and one or more gamma ray radiation detectors positioned at spaced apart locations from the neutron source. The source is periodically activated to emit controlled-duration xe2x80x9cburstsxe2x80x9d of high-energy neutrons into the earth formations surrounding the borohole. The neutrons interact with atomic nuclei of the materials in the earth formations, losing energy with each interaction until the neutrons reach the thermal energy level (defined as having a most likely energy of about 0.025 electron volts). Depending on the material composition of the earth formations proximal to the instrument, the thermal neutrons can be absorbed, or xe2x80x9ccapturedxe2x80x9d, at various rates by certain types of atomic nuclei in the earth formations. When one of these atomic nuclei captures a thermal neutron it emits a gamma ray, referred to as a xe2x80x9ccapture gamma rayxe2x80x9d.
The rate at which the capture gamma rays are emitted, with respect to the elapsed time after the end of the neutron xe2x80x9cburstxe2x80x9d depends on, among other things, the relative concentration per unit volume in the earth formations of atomic nuclei which have a relatively large tendency to absorb thermal neutrons and emit capture gamma rays in response. This tendency is referred to as the thermal neutron capture xe2x80x9ccross-sectionxe2x80x9d. A common chemical element found in earth formations having high capture cross-section atomic nuclei is chlorine. Chlorine in earth formations is usually present in the form of chloride ion in solution in connate water present in the pore spaces of some of the earth formations. Chlorine has a very high thermal neutron capture cross-section. Thus a measurement of the thermal neutron decay time (or neutron lifetime) of the earth formations in the vicinity of the wellbore can be indicative of amount of saline fluid present in the pore spaces of the earth formation. When combined with values of connate water salinity, fractional volume of pores space (xe2x80x9cporosityxe2x80x9d), and measurements of the fractional content of fine grained particles in the formation (xe2x80x9cformation shalinessxe2x80x9d) it is possible to determine the fractional fluid saturation of useful materials, such as oil or gas, present in the pore spaces of the earth formation.
It has proven difficult to determine the fractional saturation of oil or gas under certain wellbore conditions by processing the capture gamma ray measurements according to methods known in the art for determining the thermal neutron capture cross-section, xcexa3f, of the earth formation of interest. Several factors contribute to the difficulty of determining xcexa3f-using the methods known in the art. First, the well logging instrument is typically inserted into a wellbore which is filled with liquid. At the time the pulsed neutron instrument is typically used, the wellbore generally has inserted therein a steel liner or casing. The liner or casing is generally held in place by cement filling an annular space between the wellbore wall and the exterior of the liner or casing. As high energy neutrons leave the neutron source in the logging instrument, the liquid in the wellbore has the effect of rapidly moderating (or slowing down) the high energy neutrons to the thermal level because of the high concentration of hydrogen nuclei in the liquid.
In general, the relative numbers (xe2x80x9cpopulationxe2x80x9d) at any particular time after a neutron burst, of thermal neutrons in the wellbore and in the earth formations proximal to the wellbore will depend on the porosity and on the hydrogen nucleus concentration per unit volume within the earth formation. The thermal neutrons present in the wellbore and in the earth formations can be xe2x80x9ccapturedxe2x80x9d or absorbed by nuclei of various chemical elements in the wellbore and formations, at a rate which depends upon the relative concentration and on the thermal neutron capture cross-section of these elements. In wellbores and in earth formations some of the more common elements having high thermal neutron cross-sections include chlorine, hydrogen, iron, silicon, calcium, boron, and sulfur. The thermal neutron decay time or xe2x80x9cneutron lifetimexe2x80x9d, as determined from measurements of capture gamma rays made by the well logging instrument, represents combined effects of the thermal neutron capture cross-section in each of several xe2x80x9cregionsxe2x80x9d (volumes of space surrounding the logging instrument) within the wellbore as well as from the earth formations proximal to the wellbore. These regions generally include the instrument itself, the fluid in the wellbore, the steel casing, the cement, the earth formation radially proximal to the wellbore wall (which may have been infiltrated by fluid from within the wellbore), and the earth formations radially more distal from the wellbore wall (which have minimal infiltration from the fluid in the wellbore).
Determining xcexa3f-using data processing methods known in the art can be further complicated if the earth formation does not have a truly homogenous material composition on the scale of measurements made by the well logging instrument. Conditions in the earth formations subject to this difficulty can include earth formations consisting of a layered xe2x80x9csand/shalexe2x80x9d sequence wherein the layers are on the order of 3-4 inches thick, or can include the presence of a fluid transition zone such as a gas/oil or an oil/water contact in the earth formation. Other conditions can include the presence of a radial zone located within approximately 2-8 inches from the wellbore wall having a different fluid than in a radially more distal zone, this being familiar to those skilled in the art as being caused by such processes as xe2x80x9cinvasionxe2x80x9d (the previously described fluid infiltration), and gas or water xe2x80x9cconingxe2x80x9d as well as other processes known in the art.
The capture gamma ray detection rate as measured by the logging instrument will necessarily include fractional contributions from all of the regions in the vicinity of the logging instrument. Each of these regions has an indeterminate fractional contribution to the overall capture gamma ray counting rate as measured by the logging instrument, and can also have an unknown value of capture cross-section
Several processing methods are known in the art for determining the macroscopic thermal neutron capture cross-section of the formation, xcexa3f, from the measured capture gamma ray counting rates with respect to time after the end of each neutron burst (referred to as the counting rate xe2x80x9ctime spectrumxe2x80x9d or xe2x80x9cdecay spectrumxe2x80x9d). Prior art processing methods included the assumption that the thermal neutron capture cross-section of the regions within the wellbore are significantly higher than the capture cross-section of the surrounding earth formations. Limitations to these methods are described, for example, in U.S. Pat. No. 4,409,481 issued to Smith et al.
The processing method described in the Smith et al patent includes the assumption that the decay of the gamma ray counting rate with respect to time includes the effects of two and only two distinct exponential decay rates, the first caused by the materials within the wellbore and the second caused by the materials in the earth formations proximal to the wellbore. The method described in the Smith et al patent includes the assumption that the length scales of the materials in the wellbore and in the earth formation are such that the effects of neutron diffusion averages out the actual variations in capture cross-section between the various regions and therefore can be represented by some average value of thermal neutron capture cross-section. As discussed previously, several common conditions exist where this is clearly not the case. Using the processing method described in the Smith et al patent can lead to erroneous results under these conditions.
U.S. Pat. No. 5,973,321 to Schmidt describes a model based method for inversion of thermal neutron decay data. The method includes generating a data kernel which is made up of representors, or models, of potential decay components of the wellbore and of the earth formations in the vicinity of the wellbore. A thermal neutron decay spectrum is measured by a pulsed neutron instrument including a controllable source of high energy neutrons and one or more gamma ray detectors at spaced apart locations from the source. The decay spectrum measured by the instrument is inverted to determine model parameters by which the individual representors are scaled so that when combined, the scaled representors most closely match the measured decay spectrum. A potential disadvantage of model based inversion methods such as that taught by Schmidt is that the curve fitting may give different results depending upon the choice of the model used to generate the data kernel.
A similar problem is encountered in the analysis of nuclear magnetic resonance (NMR) measurements. NMR relaxation data is often multi-exponential. It consists of discrete pairs of amplitudes and times that can be the amplitudes transformed into a monotonic function of time with a zero baseline. A simple example is the inversion-recovery data for protons on a simple linear hydrocarbon chain. There are two decay rates (or times), one associated with the protons attached to CH2 groups and the other associated with the CH3 groups. The data comprises pairs of free-induction-decay (FID) amplitudes and the time between the inversion pulse and the readout pulse. The data are a monotonic increasing function of time that reach a constant value at times much longer than the largest relaxation time. By subtracting the data from this constant value, the data are transformed into a monotonic decreasing function of time.
During the early development of NMR, graphical curve stripping was used to analyze multi-exponential decay data. In this method, the data are plotted on semi-logarithmic paper, and the region in which the data decay wit only the slowest relaxation time is selected. A straight line is drawn through the data and the parameters associated with this line determine both the slowest relaxation time and its amplitude. The fitted line is subtracted from the remaining data. This strips away the slowest relaxing component and leaving the other components intact. This process is repeated until the relaxation times and amplitudes have been determined. Although tedious, this process produces satisfactory results for bi- and tri-exponential data.
With the advent of computers, non-linear fitting procedures were developed. These would solve bi- and tri-exponential data using non-negative non-linear methods. Stability of the solution is sometimes a problem when fitting both the relaxation times and the corresponding amplitudes.
In recent years, NMR relaxation data from fluid filled oil reservoir rock samples has generated much interest. Empirical relations have been established that correlate permeability with relaxation times and amplitudes. For a sample filled with a single fluid, the relaxation time distribution has features related to the pore size distribution. The relaxation time distribution can vary over several orders of magnitude from less than a millisecond to greater than one second. To obtain the distribution, one must determine the relaxation times and amplitudes for a great many exponentials. In fact, the relaxation time distribution can be considered continuous.
The solutions for this problem generally assume a series of fixed relaxation times. This linearizes the problem and the amplitudes can be fitted by a number of different methods. U.S. Pat. No. 5,517,115 to Prammer teaches the use of singular value decomposition. U.S. Pat. No. 5,291,137 to Freedman discloses the use of a maximum likelihood method. Both of these methods use exponentials as basis functions, but other basis function set can be used as well. Amplitudes whose relaxation times are close together are highly correlated, so the number of independent relaxation times is significantly smaller than the number of amplitudes used in the inversion process. The number of amplitudes usually employed varies, but typically is more than 10.
There is a need for a method of determination of the components of a relaxation time of NMR spectra that does not use any predetermined models. The present invention satisfies this need.
The present invention is a method for obtaining parameters of interest of subterranean earth formations using NMR measurements that comprise two or more exponentially decaying functions of time. A fit is made using a single exponential to the tail end of the data and the beginning of the fitting window is selected so that a product of the goodness of fit and the standard error of the fit attains a minimum. The process may be used for determination of NMR relaxation times. By subtracting the determined fit from the measurements, the process may be repeated to find additional components of the decay spectrum.